Subgroups — #27276781

Order 13568.

588 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20274867	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810355	C₂	1	2	yes	maximal
4	#20272555	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	—	2	1	no	—
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₄×#11719919	—	2	1	no	—
13	C₄×#11719919	—	2	1	no	—
14	C₄×#11719919	—	2	1	no	—
15	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
16	C₄×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
17	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
18	C₄²×Dic₅₃	—	2	1	no	—
19	#16810355	—	2	4	no	—
20	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#336	—	2	1	no	—
24	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	2	no	—
27	C₄×Dic₁₀₆	—	2	1	no	—
28	C₄×Dic₁₀₆	—	2	1	no	—
29	C₄×Dic₁₀₆	—	2	2	no	—
30	C₄×Dic₁₀₆	—	2	2	no	—
31	C₄×Dic₁₀₆	C₈	1	4	yes	complemented
32	C₄×Dic₁₀₆	D₄	1	2	yes	—
33	C₄×Dic₁₀₆	Q₈	1	2	yes	—
34	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719919	—	2	1	no	—
38	C₂×#11719919	—	2	1	no	—
39	C₂×#11719919	—	2	1	no	—
40	C₂×C₄×Dic₅₃	—	2	1	no	—
41	C₂×C₄×Dic₅₃	—	2	1	no	—
42	C₂×C₄×Dic₅₃	—	2	1	no	—
43	C₂×C₄×Dic₅₃	—	2	1	no	—
44	C₂×C₄×Dic₅₃	—	2	1	no	—
45	C₂×C₄×Dic₅₃	—	2	1	no	—
46	C₄⋊#88322	—	2	2	no	—
47	C₄⋊#88322	—	2	2	no	—
48	C₄⋊#88322	—	4	2	no	—
49	C₄⋊#88322	—	4	2	no	—
50	#13133124	—	2	1	no	—
51	#13133124	—	2	1	no	—
52	#13133124	—	2	1	no	—
53	C₂²×C₈×C₅₃	—	2	2	no	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	—	2	4	no	—
62	C₄×C₈×C₅₃	—	2	2	no	—
63	C₄×C₈×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₅₃	—	2	1	no	—
72	C₂²×Dic₅₃	—	2	1	no	—
73	C₂²×Dic₅₃	—	2	1	no	—
74	C₂²×Dic₅₃	—	2	1	no	—
75	C₂×Dic₁₀₆	—	2	1	no	—
76	C₂×Dic₁₀₆	—	2	1	no	—
77	C₂×Dic₁₀₆	—	2	1	no	—
78	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
79	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
80	C₂×Dic₁₀₆	M₁₆	1	2	yes	—
81	C₂×C₈×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	—	4	2	no	—
83	C₂×C₈×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	1	no	—
86	C₄²×C₅₃	—	2	1	no	—
87	C₄²×C₅₃	—	2	1	no	—
88	C₄²×C₅₃	—	2	1	no	—
89	C₄²×C₅₃	—	4	1	no	—
90	C₄²×C₅₃	C₂×C₈	1	2	yes	—
91	C₄²×C₅₃	M₁₆	1	2	yes	—
92	C₄²×C₅₃	SD₁₆	1	2	yes	—
93	C₄²×C₅₃	Q₁₆	1	2	yes	—
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
96	C₄²×C₅₃	#41	1	1	yes	characteristic
97	C₄²×C₅₃	#41	1	1	yes	characteristic
98	C₄⋊Dic₅₃	—	2	1	no	—
99	C₄⋊Dic₅₃	—	2	1	no	—
100	C₄⋊Dic₅₃	—	2	1	no	—
101	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
102	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
103	C₄⋊Dic₅₃	M₁₆	1	2	yes	—
104	#11719919	—	2	4	no	—
105	#11719919	—	2	4	no	—
106	#11719919	—	2	2	no	—
107	#11719919	—	2	2	no	—
108	C₄×Dic₅₃	—	2	4	no	—
109	C₄×Dic₅₃	—	2	4	no	—
110	C₄×Dic₅₃	—	2	2	no	—
111	C₄×Dic₅₃	—	2	2	no	—
112	C₄×Dic₅₃	—	4	2	no	—
113	C₄×Dic₅₃	—	4	2	no	—
114	C₄×Dic₅₃	—	4	1	no	—
115	C₄×Dic₅₃	—	4	1	no	—
116	C₂×#88322	—	4	2	no	—
117	C₂×#88322	—	4	2	no	—
118	C₂×#88322	—	4	2	no	—
119	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
131	C₂×C₄×C₅₃	#103	1	2	yes	—
132	C₂×C₄×C₅₃	#103	1	2	yes	—
133	C₂×C₄×C₅₃	#104	1	2	yes	—
134	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
146	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
148	C₂×Dic₅₃	—	2	2	no	—
149	C₂×Dic₅₃	—	2	2	no	—
150	C₂×Dic₅₃	—	2	2	no	—
151	C₂×Dic₅₃	—	2	2	no	—
152	C₂×Dic₅₃	—	2	2	no	—
153	C₂×Dic₅₃	—	2	2	no	—
154	C₂×Dic₅₃	—	4	1	no	—
155	C₂×Dic₅₃	—	4	1	no	—
156	C₂×Dic₅₃	—	4	1	no	—
157	C₂×Dic₅₃	—	4	1	no	—
158	C₂×Dic₅₃	—	4	1	no	—
159	C₂×Dic₅₃	—	4	1	no	—
160	Dic₁₀₆	—	2	2	no	—
161	Dic₁₀₆	—	2	2	no	—
162	Dic₁₀₆	—	2	2	no	—
163	Dic₁₀₆	C₄⋊C₈	1	4	yes	complemented
164	C₈×C₅₃	—	4	4	no	—
165	#88322	—	8	2	no	—
166	#8564	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	#380	1	1	yes	characteristic
168	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₂²×C₅₃	#407	1	1	yes	characteristic
170	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
171	C₂²×C₅₃	#417	1	1	yes	characteristic
172	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
173	C₂²×C₅₃	#421	1	1	yes	characteristic
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	2	no	—
177	C₄×C₅₃	—	2	2	no	—
178	C₄×C₅₃	—	4	1	no	—
179	C₄×C₅₃	Q₈⋊C₈	1	4	yes	complemented
180	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
181	C₄×C₅₃	#423	1	1	yes	characteristic
182	C₄×C₅₃	#431	1	1	yes	characteristic
183	C₄×C₅₃	#432	1	1	yes	characteristic
184	C₄×C₅₃	#460	1	1	yes	characteristic
185	C₄×C₅₃	#462	1	1	yes	characteristic
186	C₄×C₅₃	#475	1	1	yes	characteristic
187	C₄×C₅₃	#477	1	1	yes	characteristic
188	Dic₅₃	—	2	4	no	—
189	Dic₅₃	—	4	2	no	—
190	Dic₅₃	—	4	2	no	—
191	Dic₅₃	—	4	2	no	—
192	C₄²×Q₈	—	53	1	no	—
193	#1842	—	53	1	no	—
194	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
195	C₂×Q₈⋊C₈	—	53	2	no	—
196	#1279	—	53	2	no	—
197	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
198	C₂×C₅₃	#1477	1	1	yes	characteristic
199	C₂×C₅₃	#1487	1	1	yes	characteristic
200	C₂×C₅₃	#1489	1	1	yes	characteristic
201	C₂×C₅₃	#1797	1	1	yes	characteristic
202	C₂×C₅₃	#1803	1	1	yes	characteristic
203	C₂×C₅₃	#1806	1	1	yes	characteristic
204	C₂×C₄×Q₈	—	53	1	no	—
205	C₂×C₄×Q₈	—	53	1	no	—
206	C₂×C₄×Q₈	—	53	1	no	—
207	C₂×C₄⋊C₈	—	2	1	no	—
208	C₂×C₄⋊C₈	—	53	2	no	—
209	C₂×C₄⋊C₈	—	106	1	no	—
210	C₂×C₄⋊C₈	—	106	1	no	—
211	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
212	C₄×C₄⋊C₄	—	53	1	no	—
213	C₄×C₄⋊C₄	—	53	1	no	—
214	C₄×C₄⋊C₄	—	53	1	no	—
215	C₄×C₄⋊C₄	—	106	1	no	—
216	C₄×C₄⋊C₄	—	106	1	no	—
217	C₄×C₄⋊C₄	—	106	1	no	—
218	C₄³	—	106	1	no	—
219	C₄³	C₂×D₅₃	1	1	yes	characteristic
220	#336	—	2	1	no	—
221	Q₈⋊C₈	—	106	4	no	—
222	C₅₃	#8564	1	1	yes	characteristic, complemented, Sylow
223	C₂²×Q₈	—	53	1	no	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	106	2	no	—
226	C₄×Q₈	—	53	2	no	—
227	C₄×Q₈	—	53	2	no	—
228	C₄×Q₈	—	53	4	no	—
229	C₄×Q₈	—	106	2	no	—
230	C₄×Q₈	—	106	2	no	—
231	C₄×Q₈	—	106	1	no	—
232	C₄×Q₈	—	106	1	no	—
233	C₄×Q₈	—	106	2	no	—
234	C₂×C₄⋊C₄	—	53	1	no	—
235	C₂×C₄⋊C₄	—	53	1	no	—
236	C₂×C₄⋊C₄	—	53	1	no	—
237	C₂×C₄⋊C₄	—	106	1	no	—
238	C₂×C₄⋊C₄	—	106	1	no	—
239	C₂×C₄⋊C₄	—	106	1	no	—
240	C₂×C₄²	—	106	1	no	—
241	C₂×C₄²	—	106	1	no	—
242	C₂×C₄²	—	106	1	no	—
243	C₂×C₄²	—	106	1	no	—
244	C₂×C₄²	—	106	1	no	—
245	C₂×C₄²	—	106	1	no	—
246	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
249	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
250	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
251	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
253	C₄⋊C₈	—	2	4	no	—
254	C₄⋊C₈	—	106	2	no	—
255	C₄⋊C₈	—	106	2	no	—
256	C₄⋊C₈	—	212	2	no	—
257	C₄⋊C₈	—	212	2	no	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	2	2	no	—
260	#95	—	106	1	no	—
261	#95	—	106	1	no	—
262	#95	—	106	1	no	—
263	C₂×Q₈	—	53	2	no	—
264	C₂×Q₈	—	53	2	no	—
265	C₂×Q₈	—	53	2	no	—
266	C₂×Q₈	—	106	1	no	—
267	C₂×Q₈	—	106	1	no	—
268	C₂×Q₈	—	106	1	no	—
269	C₂²×C₄	—	106	1	no	—
270	C₂²×C₄	—	106	1	no	—
271	C₂²×C₄	—	106	1	no	—
272	C₂²×C₄	—	106	1	no	—
273	C₂²×C₄	#11719921	1	1	yes	characteristic
274	C₂²×C₄	#11719921	1	1	yes	characteristic
275	C₂²×C₄	#11719921	1	1	yes	characteristic
276	C₂²×C₄	#11719921	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	4	2	no	—
283	C₂×C₈	—	212	2	no	—
284	C₂×C₈	—	212	2	no	—
285	C₂×C₈	—	212	2	no	—
286	C₄⋊C₄	—	53	2	no	—
287	C₄⋊C₄	—	53	2	no	—
288	C₄⋊C₄	—	53	2	no	—
289	C₄⋊C₄	—	106	4	no	—
290	C₄⋊C₄	—	106	4	no	—
291	C₄⋊C₄	—	106	1	no	—
292	C₄⋊C₄	—	106	2	no	—
293	C₄⋊C₄	—	106	2	no	—
294	C₄⋊C₄	—	106	1	no	—
295	C₄⋊C₄	—	106	1	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	106	4	no	—
303	C₄²	—	106	4	no	—
304	C₄²	—	106	2	no	—
305	C₄²	—	106	2	no	—
306	C₄²	—	212	2	no	—
307	C₄²	—	212	2	no	—
308	C₄²	—	212	1	no	—
309	C₄²	—	212	1	no	—
310	C₄²	C₈×D₅₃	1	2	yes	—
311	C₄²	#11719912	1	2	yes	—
312	C₄²	#11719913	1	2	yes	—
313	C₄²	Dic₂₁₂	1	2	yes	—
314	C₄²	D₄×D₅₃	1	1	yes	characteristic
315	C₄²	#11719947	1	1	yes	characteristic
316	C₄²	Q₈×D₅₃	1	1	yes	characteristic
317	C₄²	#11719949	1	1	yes	characteristic
318	C₂³	C₂×#11719921	1	1	yes	characteristic
319	Q₈	—	53	4	no	—
320	Q₈	—	106	2	no	—
321	Q₈	—	106	2	no	—
322	Q₈	—	106	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	106	2	no	—
333	C₂×C₄	—	106	2	no	—
334	C₂×C₄	—	106	2	no	—
335	C₂×C₄	—	106	2	no	—
336	C₂×C₄	—	106	2	no	—
337	C₂×C₄	—	106	2	no	—
338	C₂×C₄	—	212	1	no	—
339	C₂×C₄	—	212	1	no	—
340	C₂×C₄	—	212	1	no	—
341	C₂×C₄	—	212	1	no	—
342	C₂×C₄	—	212	1	no	—
343	C₂×C₄	—	212	1	no	—
344	C₂×C₄	#13133129	1	2	yes	—
345	C₂×C₄	#13133129	1	2	yes	—
346	C₂×C₄	#13133142	1	2	yes	—
347	C₂×C₄	#13133142	1	2	yes	—
348	C₂×C₄	#13133145	1	2	yes	—
349	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
350	C₂×C₄	#13133175	1	1	yes	characteristic
351	C₂×C₄	#13133188	1	1	yes	characteristic
352	C₂×C₄	#13133192	1	1	yes	characteristic
353	C₂×C₄	#13133193	1	1	yes	characteristic
354	C₂×C₄	#13133197	1	1	yes	characteristic
355	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
356	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
357	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
358	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
359	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
360	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
361	C₂×C₄	#13133245	1	1	yes	characteristic
362	C₂×C₄	#13133250	1	1	yes	characteristic
363	C₈	—	4	4	no	—
364	C₈	—	424	2	no	—
365	C₂²	#16810480	1	1	yes	characteristic
366	C₂²	C₂×#13133129	1	1	yes	characteristic
367	C₂²	#16810586	1	1	yes	characteristic
368	C₂²	C₂×#13133142	1	1	yes	characteristic
369	C₂²	#16810625	1	1	yes	characteristic
370	C₂²	C₂×#13133145	1	1	yes	characteristic
371	C₂²	#16810641	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	2	no	—
375	C₄	—	2	2	no	—
376	C₄	—	4	1	no	—
377	C₄	—	106	4	no	—
378	C₄	—	212	2	no	—
379	C₄	—	212	2	no	—
380	C₄	—	212	2	no	—
381	C₄	#16810355	1	4	yes	complemented
382	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
383	C₄	#16810645	1	1	yes	characteristic
384	C₄	#16810675	1	1	yes	characteristic
385	C₄	#16810679	1	1	yes	characteristic
386	C₄	#16810824	1	1	yes	characteristic
387	C₄	#16810834	1	1	yes	characteristic
388	C₄	#16810909	1	1	yes	characteristic
389	C₄	#16810919	1	1	yes	characteristic
390	C₂	C₂×#16810355	1	1	yes	characteristic
391	C₂	#20273103	1	1	yes	characteristic
392	C₂	#20273145	1	1	yes	characteristic
393	C₂	#20273155	1	1	yes	characteristic
394	C₂	#20274692	1	1	yes	characteristic
395	C₂	#20274721	1	1	yes	characteristic
396	C₂	#20274736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#8564
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
31. C₄×Dic₁₀₆	C₈	semidirect
163. Dic₁₀₆	C₄⋊C₈	semidirect
179. C₄×C₅₃	Q₈⋊C₈	semidirect
222. C₅₃	#8564	semidirect
381. C₄	#16810355	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄²×Dic₁₀₆	2
#20274867	2
C₂×#16810355	2
#20272555	2
C₄×C₄⋊C₈×C₅₃	2
#8564	53